I have also (disappointingly/validatingly) thought of this and then read Tegmark. (It’s even more disappointing/validating than that, though, since as well as Tegmark, you appear to have invented Syntacticism. You even have all my arguments, like subverting the simulation hypothesis and talking about ‘closure’). However, I have one more thing to add, which may answer the problem of regularity. That one thing is what I call the ‘causality manifold’: Obviously by simulating a universe we have no causal effect upon it (if we are assuming the mathematical universe hypothesis); but it has a causal effect upon us, because it defines the results of our computation. I explore this theme somewhat in The Apparent Reality of Physics, a footnote to which mentions the problem of consistency when you have a closed loop of universes, and its putative solvability by loop unfolding / closure. Considering the ensemble of mathematical structures with the natural topology, we see that locally it’s either a graph or a manifold (almost everywhere), and it has this flow defined by the causal relations (with the flow in the opposite direction to simulation), which we can consider as being a flow of subjective probability (with some equilibrium state). Of course it contains both regular and irregular universes (hereonin RUs and IUs), because adding a delta function to a differential equation gives you, simply, a different DE (well, that’s ‘morally’ why it’s true; it’s more complicated in practice because not all mathematical structures are DEs; but any continuous mathematical structure can be continuously corrupted). IUs typically cannot simulate RUs, because any simulation is going to keep hitting the delta functions and being corrupted; RUs, on the other hand, can simulate both other RUs and IUs (a cosmic ray can turn your RU simulation into an IU simulation). Consequently, subjective probability flows from {IUs} to {RUs} much more strongly than the other way, so the equilibrium has most subjective probability on RUs. Thus, anthropics and cake for everyone :)
I should add that I haven’t yet been able to mathematically formalise the above argument, because I haven’t yet worked out the correct definitions/characterisation of the ‘causality manifold’ (which is, incidentally, not a manifold), and it’s possible that the small probability of an IU simulating a RU screws things up, and that we should (perhaps) expect to find ourselves in a Universe with some (say) Poisson-distributed degree of irregularity. Or something like that. But, at least it does allow for a mathematical universe in which anthropic experience can actually be given a probability distribution.
I have also (disappointingly/validatingly) thought of this and then read Tegmark. (It’s even more disappointing/validating than that, though, since as well as Tegmark, you appear to have invented Syntacticism. You even have all my arguments, like subverting the simulation hypothesis and talking about ‘closure’). However, I have one more thing to add, which may answer the problem of regularity. That one thing is what I call the ‘causality manifold’: Obviously by simulating a universe we have no causal effect upon it (if we are assuming the mathematical universe hypothesis); but it has a causal effect upon us, because it defines the results of our computation. I explore this theme somewhat in The Apparent Reality of Physics, a footnote to which mentions the problem of consistency when you have a closed loop of universes, and its putative solvability by loop unfolding / closure. Considering the ensemble of mathematical structures with the natural topology, we see that locally it’s either a graph or a manifold (almost everywhere), and it has this flow defined by the causal relations (with the flow in the opposite direction to simulation), which we can consider as being a flow of subjective probability (with some equilibrium state). Of course it contains both regular and irregular universes (hereonin RUs and IUs), because adding a delta function to a differential equation gives you, simply, a different DE (well, that’s ‘morally’ why it’s true; it’s more complicated in practice because not all mathematical structures are DEs; but any continuous mathematical structure can be continuously corrupted). IUs typically cannot simulate RUs, because any simulation is going to keep hitting the delta functions and being corrupted; RUs, on the other hand, can simulate both other RUs and IUs (a cosmic ray can turn your RU simulation into an IU simulation). Consequently, subjective probability flows from {IUs} to {RUs} much more strongly than the other way, so the equilibrium has most subjective probability on RUs. Thus, anthropics and cake for everyone :)
I should add that I haven’t yet been able to mathematically formalise the above argument, because I haven’t yet worked out the correct definitions/characterisation of the ‘causality manifold’ (which is, incidentally, not a manifold), and it’s possible that the small probability of an IU simulating a RU screws things up, and that we should (perhaps) expect to find ourselves in a Universe with some (say) Poisson-distributed degree of irregularity. Or something like that. But, at least it does allow for a mathematical universe in which anthropic experience can actually be given a probability distribution.